I am interested in the variational formulation of the 1D Schrodinger equation:
$i u_t- \beta u_{xx} = 0 $ and $u(x,0)=u_0(x)$ which upon integration by parts yields:
$i(u_t,v) + \beta (u_x,v_x) = 0$ and the boundary terms vanish by appropriate testing with test functions $v$. We then have the continuous sesquilinear form $b(u,v) = i(u_t,v) + \beta (u_x,v_x)$. I would like to apply the generalized Lax-Milgram on $b(\cdot,\cdot)$, but I am having trouble showing the boundedness below (coercivity) of $b(\cdot,\cdot)$. Is there a slick way to show $b(u,u) > \alpha \|u\|_{H^1}^2$?